Centralizer fusion systems of central involutions in a finite group with soluble centralizer of involutions

TL;DR

This paper classifies finite groups with central involutions whose centralizer fusion systems are nilpotent, focusing on cases with a solvable radical of odd order and simple factor groups, addressing a long-standing open problem.

Contribution

It provides a classification of centralizer fusion systems of central involutions in finite groups with soluble involution centralizers, especially when the solvable radical has odd order.

Findings

Centralizer fusion system of a central involution can be nilpotent under certain conditions.

Classification includes cases with solvable radical of odd order and simple quotient groups.

Advances understanding of the structure of finite groups with soluble involution centralizers.

Abstract

It is a long-standing open problem raised by Starostin to describe all finite groups with soluble centralizers of involutions. One can observe that if the centralizer fusion system of an involution is nilpotent, then the centralizer of that involution is soluble. In this paper, we classify the cases when the centralizer fusion system of a central involution in a finite group whose all involutions have soluble centralizers is a nilpotent fusion system. Indeed, we analyse the case when the solvable radical has odd order and the corresponding factor group is simple.